This paper presents actual labour values and production prices inherent in the structure ofthe US economy during the years 1947, 1958, 1961, 1963, 1967-70, and 1972, using a71 -industry fixed-capital model of the economy. This will be done in order to measure thedeviations of production prices from labour values and actual market prices, as well astheir relationship over time. In addition, US wage-profit curves for five of these years willbe presented.
The above project acquires its significance when viewed in the context of the emergenceof neoRicardian economic theory. Srafia's analysis of the relationship between prices anddistribution was aimed at a critique of the reigning marginalist orthodoxy (SrafEa, 1960).In this purpose, it was eminently successful. In a well-known debate carried out in a seriesof articles originating primarily from both Cambridges, the aggregate neoclassical theoryof production and distribution1 was dealt a severe—some would say fatal—blow.
This was due to two results of the Sraffian model. First, it showed that the neoclassicalconcept of capital as a 'real* factor of production whose quantity is measurable prior toexchange and distribution, was unsustainable, since the prices used to measure the valueof capital goods depend on distribution. Second, by raising the possibility of reswitcbingof techniques as the rate of profit (and distribution) varies, it showed that the results ofmarginal productivity theory—inverse relation between capital intensity and the rate ofprofit; die rate of profit as the scarcity price or efficient allocator of capital—weretheoretically untenable.
The same analysis, however, contained an implicit criticism of the Marxian labourtheory of value. The criticism soon turned explicit (Steedman, 1977). The Srafia systemwas said to make Marx's value analysis (and the subsequent transformation to prices ofproduction) redundant. Moreover, the economic structure which emerged from a labour-value analysis, it was argued, did not necessarily match the observed structure in terms of
Manuscript received 1 August 1986; final version received 30 August 1988."California State University, Los Angeles. This paper is based on portions of my doctoral dissertation
(Ochoa, 1984). I would like to thank Anwar Shaikh, my principal dissertation advisor, for his guidance anddiligent support.
1 See Ferguson (1969) for a comprehensive statement of the neoclassical aggregate production anddistribution theory. As was argued by Samuelson and others during the debate between the two Cambridges,reswitching did not affect the internal consistency of neoclassical theory in its full generality, in the Walrasianmanner developed by Arrow and Hahn (1971), among others. But for a 'classical' critique of even this generalform, see Dumenil and Levy (1985).
prices. Specifically, the rate of profit in value terms did not equal the rate of profit in termsof production prices, the sum of prices did not equal the sum of values, and the sum ofprofits did not equal the sum of surplus value. Moreover, their deviations were notsystematic, but depended on the vagaries of distribution. In fact, the adoption of newtechniques did not follow a clear pattern in terms of value quantities (such as a risingorganic composition of capital) but again depended on distribution and the ensuing pricechanges, since the decisions of individual capitalists are based on the observable pricesalone.
The possibility of reswitching was once again invoked as an illustration of adjustmentsin the system in response to price changes which are independent of any changes in valuemagnitudes. This argument relies on the logical or algebraic possibility of magnitudesof the technical coefficients such that price-value deviations will be very large, andwage-profit rate curves will have the shapes required for reswitching of techniques, ordistributional effects so large that value and price analyses will lead to differentconclusions.
There is no question that these are mathematical possibilities of the model. Whetherthey are real possibilities of any actual economy, however, is another, more questionablematter. This paper will attempt to shed some light on this last question.
2. The Sraffian critique of Marxian value theory
The reswitching phenomenon in a Sraffian system can be briefly characterised asfollows. Given a capitalist economy which produces n commodities using n single-product processes, the production-determined equilibrium prices which are formeddepend on the distribution of the net product between wages and profits. Moreover, it ispossible to construct two viable matrices of technical coefficients differing only in onecolumn (set of sectoral inputs) such that one of the matrices (technologies) is moreprofitable than the other at high average rates of profit, less profitable at intermediate ratesof profit, and once again more profitable at very low rates of profit. This is describedas a switch out of, and a re-switch back to, the original technology as it assumes its range ofvalues.
The criticism of Marxian value theory implicit here is that, since changes in distributioncan reverse the ranking of techniques in price terms, their ranking according to whichproduces the given use-value at the lowest value (i.e., embodied labour time) is irrelevantto economic decision-making by individual capitals, and hence to the aggregate of thosedecisions. It follows that aggregate value analysis ('capital in general') cannot predict anydynamic tendencies operating at the price level (i.e., the laws of motion of capitalistproduction as a whole).
If the examples constructed to illustrate reswitching are representative of thetechniques of production present in real economies, then this criticism has powerfulpractical force. However, if a careful study of all the available data on actual technologymatrices which have existed over time yields no such cases, then the likelihood increasesthat reswitching is a mathematical curiosum of no real significance, rather like negativeprices. By examining the technologies of the US economy for the period for which data areavailable (1947-1972), and the associated production prices and wage-profit rate curves,this paper seeks to make a contribution to this necessary empirical determination of thecharacter of the reswitching phenomenon.
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Values, prices, and wage-profit cuurves im the US economy 415
On a more general level, the neoRicardian criticism of Marxian value analysis rests onan emphasis on the quantitative divergence of prices from values. In this criticism, pricesof production are identified with market prices; in other words, the economy is assumed tobe in equilibrium. No real economy, however, is ever in equilibrium. The distinctionbetween actual prices (market prices) and equilibrium prices (prices of production forall but the marginalists) is therefore crucial. The argument usually made against valueanalysis is that, since individual capitalists base their decisions on observed prices, valuequantities which differ markedly from prices yield incorrect predictions of capitalists'behaviour.
A number of methodological objections which are independent of the magnitude ofprice-value deviations have elsewhere been made to this argument (Steedman, 1981). Inthis paper, however, we shall focus on two narrower points on which empirical evidencecan be brought to bear. First, the divergence of theoretical predictions and actualbehaviour of capitalists is equally likely to occur when comparing prices of production andmarket prices. It would seem in fact quite likely that values and prices of production arequantitatively closer to each other than either is to market prices, which would imply thatvalue analysis is as good a practical tool of analysis of real economies as equilibrium-priceanalysis. Empirical confirmation of this hypothesis would be highly significant.
Second, one of the purposes of the value category in Marx's analysis is to show that theamount of socially necessary labour time required to produce commodities is the essentialreality behind the formation of prices, and that therefore changes in the former will be themain long-run determinants of changes in the latter. This is likewise an empiricallytestable hypothesis.
3. Price systems
We now present labour values and three different types of price systems for the USeconomy for the years 1947,1958,1961,1963,1967,1968,1969,1970 and 1972.
3.1. Labour valuesLabour values are here assumed to be identical to total (direct and indirect) labourrequirements for unit of output. The approach followed in dealing with the problem ofheterogeneous labour is discussed in the appendix. In addition, we are ignoring dynamiceffects such as rapidly changing technology and changing patterns of demand. The formereffect would make the socially-necessary labour time—as determined in the sphere ofproduction by the state-of-the-art technology—different from the (average) total labourrequirements for the industry as a whole. The latter effect would imply a disjunctionbetween the two senses of socially-necessary labour time (the second sense being theamount of social labour time which 'society' is willing to allocate to the production of agiven good, as evidenced by the level of demand). Assuming these effects away is anabstraction comparable to assuming that the rate of profit is uniform across industries.Moreover, it should be possible to infer the real importance of these effects from theempirical results.
The system of 71 linear equations which define labour values is:
v=ao+v(A+D) (1)
where a,, is the homogeneous-labour-coefficients row vector, A is the matrix of inputcoefficients whose elements a- represent the amount of good i used by industry j , and D is
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the physical capital depreciation matrix (see the appendix for a discussion of data sourcesand methods). The solution to (1) is given by
v=ao(I-A-Dy1 (2)
Equation (2) defines the row vector v as the amount of labour required directly andindirectly to produce one unit of sectoral output. That physical unit is a 'market-dollar'sworth', given the way a^ A and D are defined. When we use their deflated versions, theunit is a constant (1972) market-dollar's worth, which is then a constant measure ofphysical output. The vector v thus has the units of labour+time per physical unit ofoutput.
3.2. Direct pricesWe define the row vector of direct prices d, following Shaikh (1977), as the set of pricesdirectly proportional to labour values, where the constant of proportionality relates themoney unit to a unit of labour time (i.e. worker-hours/dollar). We define the money unitby requiring that the sum of sectoral outputs at direct prices equal the sum of sectoraloutput at market prices. In vector notation,
dq = mq (3)
where the vector of market prices m is the unit vector, since q is measured in marketprices. The proportionality constant fi (the value of money) is therefore given by:
vq W
3.3. Marxian prices of productionMarx defines prices of production as the sum of costs plus an intersectorally uniform rateof profit on capital advanced. These magnitudes are seen to be the centres of gravity of thecontinually-fluctuating market prices. By capital advanced, we mean the capital investedin plant and equipment (fixed capital), plus the accumulated investment in inventories ofmaterials, plus the stock of money necessary to pay out wages. The level of materialinventories and wages fund is related to their flows by the turnover time of circulatingcapital tj in the/th industry where; = 1 , . . .71.1fwehavetheflowsperyear,andtheturnovertime in (fractions of) years, then the necessary stocks are simply (annual flows x turnovertime). This would give us the stock levels necessary to produce one year's output. Dividingthrough by the output level, we obtain the stocks required per unit of output.
In order to specify the wage, we must include the real counterpart of the value of labourpower. This is given by the column vector b', which is the real wage basket per unit ofhomogeneous labour time. The expression for p, the Marxian prices of production vector,is then as given below:
p=p(b\+A+D+ <g>)+np[K+(A+b'aJ<t>] (5)
where b'a,, is the matrix of wage-good inputs, <g> is the diagonal matrix of indirecttax coefficients (see the appendix), and n is the uniform rate of profit. Let A+=(b'ao+i4+D+<^>) (i.e. total costs) and K+[K+(A + b'aJ<t>) (total capitaladvanced). Then (5) reads
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Values, prices, and wage-profit curves in the US economy 417
or/-^+)-1 (6)
Equation (6) is the eigenvalue problem for the matrix K+(I—A+)"').The economically meaningful solution requires p to be a strictly positive, real vector.
If we assume that JC+(I—A*)~x is an indecomposable matrix—and we know it is non-negative—then the Perron-Frobenius theorem ensures that the only such eigenvector isthe one associated with the largest eigenvalue (1/jOnm, (to which corresponds the lowest n).
Since p is an eigenvector, it is denned up to a constant. In other words, (6) only definesa set of relative prices. To set the price level, we need a normalisation condition similarto (3):
p q = m q (7)
Let p* and p be the unnormalised and normalised eigenvectors, respectively. Then wedefine the normalisation constant /? such that p=/?p*. We can then rewrite (7) as
0p*q = mq (8)
which implies
Therefore,
mq
Summing up, Marxian prices of production are given by solving for the eigenvectorassociated with the maximum eigenvalue of the system of equations (6), and thennormalising this eigenvector according to (10).
3.4. Sraffian prices of productionThe generalised price system developed by Sraffa in Part II of his book (Sraffa, 1960)treats fixed capital as a joint product. Consider the following price system with jointproducts:
(l+;r)s/C+8^+a0to=sB (11)
K is the capital stock (which becomes a flow in a joint-product model) and A is the flowof circulating capital (materials), B is the matrix of joint products, eo is the scalar wage, ands is the price vector.
We shall present the conditions under which the Sraffian joint-product model of fixedcapital reduces to a standard depreciation model. Define B=B—I. Then B=6+I,so thatthe joint-product matrix B is the sum of the identity matrix (i.e. the unit output of eachindustry) and the matrix 6, now interpreted to be the used-machinery coefficients (i.e.capital as a joint product). Then the above equation can be rewritten as follows:
a (12)
The matrix (K—6) represents the difference between the stock of capital going into theproduction process and the stock of capital which emerges out of it; in other words, the
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scrappage matrix. This is precisely the series which we use to construct the matrix D in ourprice systems, so the above equation is none other than
(13)
This approach to fixed-capital models is discussed and criticised by Varri (1980)because the scrappage matrix (K—B) cannot be defined independently of prices. Never-theless, in any discussion of actual economies and calculations using empirically-obtainedcoefficients, this is all that is available. What the above discussion shows is that a joint-production treatment of fixed capital is equivalent to the standard treatment provided weassume that the price of old machines (and hence the decision to scrap) is not sensitive tochanges in the distribution of income.
Even if this is the case, however, Sraffa's model is still different from Marx's, since hecomputes the profit rate on fixed capital advanced only. He also does not have a concept ofthe value of labour power, so the distribution of the surplus product between wages andprofits is left open as a degree of freedom of the system. We can solve for the resultantsystem as follows:
s = coa0(/-/4-.D-7t/Cr1 (14)
This is a system of n linear equations and n + 2 unknowns: s, a>, and n. We also have ourusual normalisation condition
sq=mq (15)
Combining (14) and (15) and specifying the level of n, we get:
coa0(I—A—D—nK)~lq=mq
or
mq
Solving (16), we obtain the money wage, which we can use in (14) to obtain theprice vector s. This can be done for a number of values of n in the range (f?,R), where Ris the inverse of the maximal eigenvalue of the system given below, which is (13) with60=0:
sR=sR(A-D)+RsRK (17)
4. Sectoral price-value deviations
Following the definitions presented above, labour values v, direct prices d, and Marxianproduction prices p for the nine years studied were calculated. This section of the paperwill present results based on those basic computations.
In order to measure the extent of the pair-wise cross-sectional deviation between directprices, Marxian prices of production and market prices in a real economy, we developedthe following statistics (here illustrated for the production price-direct price case ofTable 1):
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Values, prices , and wage-profit curves in the U S e c o n o m y 419
Mean Absolute Deviation (%): (18)
MAD(p,d)=(l/n) I !?CZd (100)• d,
Mean Absolute Weighted Deviation (%): (19)
In—dj qMAWD(p,d) = I — • — (100)
•' d,. ?q,
Normalised Vector Distance (%): (20)
NVD(p,d)= - —-—(100)
Both MAD and MAWD are the mean of the absolute value of the fractional deviationsof one set of prices from another. The NVD measures the vector distance between thepriced output vectors using two sets of prices as a fraction of the vector length of one of theformer.
We also computed the cross-sectional correlation coefficient, which we report squaredas R2. In order to minimise spurious correlation—since prices and values must be corre-lated as p,q,- and v,<j, to have any variation in the market-price data—we computed R2 onthe logged data points. Nevertheless, as has been shown elsewhere (Ochoa, 1984; Petrovic,1987), measures of covariance such as R2 are not the proper statistics to assess the cross-sectional relation between alternative price systems. Rather, measures of deviation such asthose presented above should be used.
The choice of numeraire should be the money unit, since that is how the exchange valueof goods is actually measured. But the money unit's exchange value is in effect given by theequation of purchase price of total output to the quantity of total output. In effect, the unitof value is represented by a quantity vector with the proportions of the gross outputvector, whose price is one dollar:
q = Q/mQ = Q/r (21)
where Q=gross output vector; p = computed-price vector; T— scalar market price ofoutput vector=mOj q=composite numeraire (vector); m=market-price vector.
Then the normalisation condition is
p q = l orpQ/mQ = l orpQ = mQ
which is in fact the condition adopted in this study.To measure the extent to which individual values determine the behaviour of produc-
tion prices over time, we also performed 71 time-series linear regressions of values onMarxian production prices, using constant-dollar market prices as a pseudo-quantitymeasure. The associated correlation coefficients were averaged and squared to obtain ageneral measure of explained variation which is dimensionally comparable to the cross-sectional R2. (The reason we squared after averaging was that we did not want to takecredit for negative correlations over time. Using this procedure, the latter actually reducethe magnitude of the /?2-like time-series statistic.) Unlike in the case of cross-sectionaldeviations, here it is legitimate to use R2 as a measure of correlation, because all three sets
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of prices (market, direct, and production prices) are varying over time independently ofoutput levels.1
Calculated values for all these measures are presented in Table I.2 These results showclearly that labour values have a very high degree of cross-sectional correlation with pricesof production. This admittedly inconclusive result is similar to results reported elsewhere(Wolff, 1979; Petrovic, 1987). More significantly, the average correlation over time is alsoquite high: approximately 93% of the variation in individual prices of production overtime is due to changes in the underlying labour values, suggesting of all things a Ricardian93% labour theory of value. In addition, average price-value deviations—whetherweighted or unweighted—are quite small: around 17%. The 'transformation problem',therefore, appears to be of limited empirical significance.
1 The following set of relations were estimated:
0j i = l , . . . , 71; £(iO=0. (i)
The relation above refers to unit prices, where the physical unit remains constant throughout the time-series. Our actual results use a current market dollar's worth as the physical unit; we needed to convert it to aconstant (1972) market dollar's worth to obtain prices for an unchanging physical unit.
Our actual results were
(ii)We could not regress £>,{t) on Af,<r) without spurious correlation, due to the appearance of q,<0 on both
sides. So we can divide through by Af/t):Pit) d,<t)q,<t) d,<f) M/jt) t
M,(t) m,<»)q,<0 m / i ) ' M,(t) (u>)This is the form in which we have presented direct prices: djmt. While this eliminates q/t) from both
expressions, it turns market prices into a constant (one), which means there is no variation left to correlate.Equation (i) is equally valid when divided through by a constant (m,(1972)), with suitable redefinition of a,and a,.
01,(0
which gives prices per 1972 market dollar's worth of sectoral output. Butd/Q d,<f) mfr)
m,<1972) m.<0 m.0972)Since m/r)/m,{1972) is nothing but the price-index e,{')>wc may write (iv) as
Equation (v) is equivalent to (i). The price-index e,{r) is clearly a time-series of market prices; but the left-hand side appears to t~ra in the same variable. A glance at (iv), however, quickly dispels mat impression: thee,{r) is mere precisely to eliminate the influence of m/r) in the denominator of the left-hand side of (v).
1 Empirical results not presented in this paper are available to readers from the author upon request.
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marketprices (one dollar's worth) as the quantity unit of each sector, so that all computed pricescluster around unity. However, just as the mean value of a distribution is irrelevant whenwe measure its coefficient of variation (standard deviation/mean), so are the absolutevalues of the percentage deviation of one set of prices from another (see equations (18) and(19) above) independent of the scaling factor used. Whether we have actual p,- or p,-/m- aswe do, the terms being summed in equations (18) and (19) would be the same:
Moreover, in the case of our NVD measure, market prices are eliminated from theexpression of computed prices because we use the value of the full sectoral output:
The results presented above become even stronger when we consider the actual extentof the deviation between production prices and market prices, shown in Table 2.
A comparison of the results of Tables 1 and 2 shows that prices of production are nearlyas far away from market prices as they are from direct prices. Since there is no reason toexpect that these deviations are correlated, and from theoretical considerations, we wouldexpect that the deviation between direct and market prices would be more than that ofproduction prices from market prices, but substantially less than the sum of the twodeviations. It was surprising to find that, in fact, the observed direct price-market pricedeviations are smaller than the production price-market price deviations (with theexception of the MAD in one year) as shown in Table 3.
While both MAD and MAWD measures are smaller between direct and market pricesthan between production and market prices, their Rh are smaller as well. This suggests
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the possibility of systematic bias on the computation of production prices, leaving themfarther away from market prices, even though more covariant with the latter than directprices (we see here the usefulness of calculating R2s as a means of estimating relativedegrees of correlation, even though the absolute values are biased upward by the spuriouscorrelation of output levels). The likely source of bias in the production-price compu-tation is the capital stock series. The capital stock series issued by the Bureau of IndustrialEconomics of the US Department of Commerce (1983) is based on the 'perpetual inven-tory' method. The latter is essentially an integration of past investment flows coupledwith a probabilistic scrappage function. This function uses a distribution centred aroundestimated asset lives. The results of the integration are very sensitive to the asset livesused, and the values used are known to be unreliable estimates.
Over time, prices of production account for somewhat more of the variation in marketprices than do direct prices (Rh equal 0-760 and 0-754, respectively), but the improve-ment is quite small compared to the amount of variation left unaccounted for. Thissuggests that labour values are the dominant deterministic influence over market prices,with the distribution effects of production prices playing a far less important role. More-over, the distribution effects—the improvement obtained by the 'transformation' fromvalues to prices of production—are themselves a fraction of the 'noise' component ofmarket prices (the stochastic disequilibrium effects).
These results suggest that the labour theory of value is not only a powerful methodologyfor a critical understanding of the social relations of production in capitalism. It alsoappears that labour values are quantitatively dominant influences in the formation ofmarket prices. The conceptual equalisation of the rate of profit which defines prices ofproduction as the centre of gravity of market prices thus provides only a marginally betterapproximation to the latter than labour values themselves.
5. Price-value deviations and wage shares
The results presented above relative to price-value deviations are so startling that therehas been and will be a tendency to minimise their importance by attributing them to thehigh wage shares in the US economy. In the limit, when profits go to zero and wages takeall the net product, prices of production would become identical to values. Hence, if thedistribution of the net product between profits and wages were close to this limit, prices ofproduction would be close to values regardless of the manner in which the structure of theeconomy would transform values.
In order to investigate this possibility, we should vary the distribution of the netproduct between wages and profits, derive the corresponding prices of production, andobserve the price-value deviations which ensue. To do this, we begin by replacing the realwage vector b' with a variable scalar wage u>. From (5) we have:
p=pb'eo+p(A + D+ <g>)+npb'ao<t> +7cp(K+A<t>)
But pb' is a scalar (the production-price of the real wage vector); so we designate it co.Solving the above equation for p we thus obtain:
p=(oa0(I+n<t>)[I-A-D- <g> -n(K+A<t>)]~1
This is no longer an eigenvalue problem, but a linear system with two degrees offreedom. Combining the above expression with our usual normalisation condition leavesone degree of freedom. By setting n equal to values from 0 to R, we can obtain the
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desired prices of production. The resultant price-value deviations for 1967 (a typical year)are shown in Table 4. It should be noted that it is the presence of indirect taxes as acomponent of costs in the price equation (5) that prevents the measures of deviationbetween production prices and direct prices from vanishing when n=0.
These results show several things. First, the wage share of income in the US was by nomeans as high as is usually implied: the year 1967 shows an average wage share of income(after depreciation and indirect business taxes) of 52-2%. Second, the measures ofdeviation predictably become larger as the wage share is reduced, but they increase onlymoderately: even when the wage share drops to zero, the MAD is only 34-4%. Thecross-sectional R2s average 0-939 even for this extreme case. Similar results were obtainedfor the remaining eight years in this study. We conclude that the remarkably close corre-spondence between direct prices and prices of production is a feature of the actual USeconomy which is not sensitive to the level of the wage share.
6. Empirical wage-profit cuirves
Implicit in Marxian theory is the belief that an analysis of technical change carried out invalue terms is not qualitatively modified in actual capitalist economies owing to thedeviations of prices from values. It is the essence of the reswitching argument, however,that such 'distributional' sources of variation can generate phenomena all of their own,and thus not only destroy the neoclassical parable, but also make any analysis of choice oftechnique in terms of labour values inconsistent with the results in terms of prices (ofproduction).1
There is no question but that it is possible to construct numerical examples of 2- orw-sector economies which will exhibit reswitching properties. The question emerges,however, as to whether such cases are merely logical curiosities or real possibilities foractual economic systems.
Unfortunately, we do not possess the information about all available techniques in eachsector of production which would be necessary to calculate wage-profit frontiers. Using
1 In this regard, see Parys (1982).
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our 'Sraffian' prices of production,1 we can generate wage—profit curves for each year'seconomy. While these are not the wage-profit frontiers which are invoked in the reswitch-ing argument, they are the only computable wage curves, and provide suggestive evidenceon the empirical relevance of reswitching. To obtain them, we simply solve equations (14)and (16) above for values of n firom <p to R. By using coefficients and output levels measuredin constant dollars, we obtain values of the money wage which are directly comparablefrom year to year. The wages are, in effect, deflated.2
The results of this procedure for the years 1947,1958,1963,1967,1972 are plotted inFig. 1. This shows the wage-profit curves of the US economy for the five years in question,spanning a period of 25 years. The most striking feature of all of them is how nearly linearthey are: they exhibit very slight convexity, with extremely small Sviggles' close to the re-intercept.3 This is due to the remarkable closeness of labour values to prices of productionfor the US economy.
1 We chose to use what we call Sraffian prices because these most closely approximate the terms of thereswitching debate. Marxian prices of production, however, yielded nearly identical curves.
2 They are not, however, deflated by the equivalent of the GNP deflator. Rather, by what we might call'gross output deflator', in the input-output sense of the term.
1 Our description of the wage-profit curves as nearly linear can be justified by a simple quantitativemeasure: a stepwise linear regression of si on powers of n using our calculated points. For 1967—a typicalyear— we observed levels of explained sum of squares of deviations shown below (sum of squares explained byeach variable when entered in the order given):
Due to Sum of squares
jr1
n>
ResidualTotal
9-6824450-10890600009750-00000900000019-792335
Alternatively, when we regress to on n alone (i.e., approximate the wage-profit curve with a straight line) weobtain for 1967 an IP of 0-989. Similar results hold for the other years.
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(Standard deviation= 121; mean =102-4; standard deviation/mean = 012).
There is also a strong trend toward higher net product per worker, as evidenced by therising a>-intercept. The output-capital ratio (or maximum rate of profit R) fell dramati-cally from 1947 to 1958; after which it rose steadily until 1967. Between then and 1972 itfell once again.1 In other words, the character of technical change from 1947 to 1958 andfrom 1967 to 1972 follows Marx's characterisation (falling unit values and falling R); from1958 to 1967, however, the wage-profit curve moved strictly outward, so that the newtechniques were unambiguously more profitable regardless of the wage rate. This impliesthat the new techniques were both capital- and labour-saving in this period.
Another noteworthy feature is the actual position of the economy during these years, asshown by the small circles on the curves. Notice that in every single instance, the newcurve would have yielded a higher rate of profit if the wage rate had remained unchangedfrom one time period to the next. The higher wage rates alone were responsible for thefall in the profit rate in the two periods when this happened; in the other two periods,the rise in the wage rate was more than compensated for by the outward expansion of thewage-profit curve (specifically, by the rise in R).
In fact, the only periods where the actual n fell are precisely those for which theassociated R fell; also, the rise in the wage rate closely mirrors the rise in output perworker. Both of these characteristics are consequences of the fact that the rate of surplusvalue remained relatively steady throughout this entire period, as shown in Table 5.
7. Conclusion
This paper has shown that labour-values and prices of production for the US economy inthe post World War II period were remarkably close to each other as well as to marketprices. The scale of the errors to be expected in the available data suggests that little if anyaccuracy is to be gained by calculating prices of production, so that either value or market-price series should be adequate in studying the behaviour of the economy in the aggregateand over time. This is a startling empirical postcript to the long-standing debate onthe 'transformation problem* which now appears to involve relatively insignificantmagnitudes in real economies.
Equally remarkably, the wage-profit curves implicit in the input-output coefficients forthe period all were very nearly linear. Moreover, this latter result does not depend on thecomposition of output or the weighting scheme used to homogenise labour inputs. As iswell known, linear wage—profit curves are obtained when Sraffa's standard product isused as the numeraire, so if the US economy was producing in near-standard proportions,this could explain our results. However, when we calculated the standard product pro-portions and compared them to the actual ones, we found them drastically different: for1967—a typical year—the MAD between standard and actual products was 386 %, and the
1 Correcting for capacity utilisation does not alter these results (Ochoa, 1984).
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R2 0-037. The weighting scheme used for labour inputs was uncorrelated with capitalintensity; moreover, the results with unweighted labour inputs are not significantly differ-ent (Ochoa, 1984).
While the wage-profit curves calculated cannot be brought directly to bear on thetheoretical reswitching debate, the fact remains that over a period of 25 years the economyhas exhibited wage—profit curves (i.e. techniques) which are a far cry from the apparentlyunlikely shapes required for reswitching and capital, reversing to occur. While thepresence of heterogeneous capital goods and fixed proportions dealt fatal blows to theneoclassical concept of aggregate physical capital, the near-linearity of actual wage-profit curves appears to support the labour theory of value as a powerful practical tool toanalyse and understand the global character of production and growth in capitalisteconomies.
BibliographyArrow, K. J. and Hahn, F. H. 1971. General Competitive Analysis, San Francisco, Holden-Day;
Edinburgh, Oliver and BoydCarter, A. 1970. Structural Change in the U.S. Economy, Cambridge, Mass., Harvard University
PressDumenil, G. and Levy, D. 1985. The classical and the neoclassical: a rejoinder to Hahn, Cambridge
Journal of Economics, vol. 9, no. 4Ferguson, C. E. 1969. The Neoclassical Theory of Production and Distribution, Cambridge, CUPMarx, K. 1967. Capital, 3 volumes, New York, International PublishersMarzi, G. and Varri, P. 1977. Variaziom di Produttivita nell'Economia Italiana: 1959-1967,
Bologna, II MulinoOchoa, E. 1984. Labor Values and Prices of Production: An Interindustry Study of the U.S. Economy,
1947-1972, Ann Arbor, Mich., University MicrofilmsParys, W. 1982. The deviation of prices from labor values, American Economic Review, DecemberPasinetti, L. 1977. Lectures on the Theory of Production, New York, Columbia University PressPetrovic, P. 1987. The deviation of production prices from labour values: some methodology and
empirical evidence, Cambridge Journal of Economics, vol. 11, no. 3Research Data Associates, 1982. Draft Documentation for Interindustry Data Tape, Worcester,
Mass.Schneider, H. 1964. Recent Advances in Matrix Theory, Madison, UWPShaikh, A. 1977. Marx's Theory of Value and the Transformation Problem, in The Subtle Anatomy
of Capitalism, edited by J. Schwartz, Santa Monica, Cal., Goodyear Publ.Shaikh, A. 1983. 'Two different standard systems', mimeo, New School for Social ResearchSraffa, P. 1960. Production of Commodities by Means of Commodities, Cambridge, CUPSteedman, 1.1977. Marx after Sraffa, London, New Left BooksSteedman, I. (ed.) 1981. The Value Controversy, London, New Left BooksSweezy, P. 1970. The Theory of Capitalist Development, New York, Monthly Review PressUS Department of Commerce, 1970. 'The Input-Output Structure of the U.S. Economy: 1947',
staff paper, Washington, DCUS Department of Commerce, 1968. 'Input-Output Transactions: 1961', Bureau of Economic
Analysis Staff Paper No. 16, Washington, DCUS Department of Commerce, 1975. 'Summary Input-Output Tables of the U.S. Economy: 1968,
1969,1970', BEA Staff Paper No. 27, Washington, DCUS Department of Commerce, Survey of Current Business, various years. Nov. 1964, Nov. 1969,
Nov. 1973, Feb. 1974, Feb. 1979, March 1979 issues, Washington, DCUS Department of Commerce, 1981. GNP and components (14) by industry, 1948-79; 1980
benchmark, Magnetic tape, Washington, DCU S Department of Commerce, 1983. Bureau of Industrial Economics Capital Stock Data Tape,
Washington, DCUS Department of Labor, 1979. Capital Stock Estimates for Input-Output Industries: Methods and
Data, Washington, DC
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Values, prices, and wage-profit curves in the US economy 427
US Department of Labor, 1973. Employment and Earnings: U.S. 1909-1971, Washington, DCUS Department of Labor (various years). Annual Supplement to Producer Prices and Price Indexes,
Washington, DCUS Department of Labor, 1979. Time Series Data for Input-Output Industries, Washington, DCVarri, P. 1980. Prices, Rate of Profit and Life of Machines in Sraffa's Fixed-Capital Model, in
Essays on the Theory of Joint Production, edited by L. Pasinetti, New York, Columbia UniversityPress
Wolff, E. 1979. The rate of surplus value, the organic composition, and general rate of profit in theUS economy, 1947-1967, American Economic Review, March
Appendix; data sourcesThis is an account of the sources and methods used to perform the calculations reported in thispaper. For further details, see Ochoa (1984). We have, from the US Department of Commerce, A,the matrix of input-output coefficients for the US economy for the years mentioned above, at variouslevels of disaggregation. We worked at the 82-industry level, aggregated to 71 industries, in order tomatch the capital-stock data available. Unlike the technical coefficients matrix used in Sraffianmodels, these matrices show the input requirements in dollars per dollar of output (the Sraffiantechnical-coefficients matrix shows the interindustry requirements in real terms). From the samesource we have q, the dollar value of output-by-industry vector, at 82-order.
We have, from the US Department of Labor, the direct-labour requirements, in worker hours,per dollar of output, at 82-order, for the years listed above. Since these labour times are hetero-geneous (different skills and intensities), we used the relative wage structure in these industries,obtained from the same source, to reduce skilled or more intense labour to unskilled labour ofminimum intensity, as exemplified by the lowest-wage sector. We thereby assume that the labourmarkets have no significant barriers to entry, so that the relative wage structure is a good measure ofthe relative values of heterogeneous labour powers and the rate of surplus value is uniform acrossindustries. We call the resultant reduced-labour requirements vector, a,,.
From the US Department of Commerce (1983), we have the vector h, the dollar value of grosscapital stock in current dollars for each industry. Since this capital stock is heterogeneous, we usedasset weights to to disaggregate Is in G, the gross capital-stock matrix, whose elements Gn show thedollar value of the stock of the ith capital good in the/th industry. Dividing each column; of G# by q,,we obtain the matrix of capital-stock coefficients K^. The set of asset weights were derived byassuming that the stock of every capital good has a uniform age distribution, so that the fractionscrapped each year is the inverse of the asset lifetime (given in US Dept. of Commerce, 1979). Wethen assumed further that the composition of the capital stock for each industry, in terms of the71-commodity structure we are using, changes slowly over time. It follows that there is a straight-forward relation between the composition of gross investment—which is given for 1963,1967, and1972—and the composition of the capital stocks. The composition of net investment will be the sameas that of replacement investment, and clearly the same as their stun (gross investment). By assump-tion, the following relation holds between replacement investment (depreciation) and stock of the ithgood in the/th industry:
"J-̂ -'Ni-V'* ( A 1 )
where 1, is the lifetime of the ith asset. The relative composition of the capital stock w^ is thereforegiven by
(A2)
and we obtain K from k using the expression below:
K—wJij (A3)
We also have 71 -order depreciation vectors d,. (gross discards) of capital stock for the relevant years(US Dept. of Labor, 1980). These depreciation levels are straightforwardly disaggregated—givenour assumption of constant composition—as follows:
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where H$ is the known gross investment matrix. This also allows us to compute wd- as outlined above.We proceed to discuss the conditions under which it is appropriate to use gross discards as a
measure of the physical wear-and-tear of capital goods. Assume a uniform distribution of ages foreach kind of asset held by each industry. If this assumption is reasonable, then each year's grossdiscards of equipment by each industry will equal total physical wear-and-tear for all assets held bythat industry. The latter quantity (which is defined prior to distribution) need not be obtained byassuming linear depreciation: in fact, in the statistics which we used, it is not. This proposition can beshown to be true as follows. Let d^ i = 1 , . . . , « be the fractional physical wear-and-tear which occursfor a given kind of asset in a given kind of industry in year i of this asset's life (which is n years). The d,may all be different, but we require that ? d,= 1 (i.e. the machine transfers its value to the productexactly over its life). Given our assumption that we have a uniform distribution of ages, the fractionof the total assets of this kind in this industry of age i years i s / = 1/n, for all i. Now we show that thephysical amount of gross discards equals the total amount of physical wear-and-tear on all the assets(of this kind in this industry).
1. Every year, 1/nofall the assets are discarded (as given by our gross-discards series).2. Also every year, the total amount of physical wear-and-tear is given by
Hence, the physical amount of gross discards equals the physical amount of wear-and-tear,assuming a uniform distribution of vintages for this kind of asset, but without having to assumelinear wear-and-tear.
It might be questioned how we could define wear-and-tear in purely physical terms, or rathergross discards. This was done the same way we measured all other physical quantities in this project:by making one current-market-dollar's-worth the physical unit of measurement. Hence the grossdiscards were measured at replacement cost in current dollars. Then depreciation is given as sD(i.e. not before distribution).
To obtain the real unit wage vector b, we use the sectoral proportions in the PersonalConsumption Expenditures component of final demand, which is part of the input-output datadeveloped by the Department of Commerce. By multiplying it by the wage rate in current dollars ofthe lowest-wage sector, we obtain the dollar amount of each consumer good required per unit ofreduced labour.
The treatment of indirect business taxes in this study assumes that they represent a cost like anyother in the formation of a uniform rate of profit. We therefore compiled a vector of indirect-taxcoefficients, g, from US Department of Commerce (1981).
The turnover-time vector t was estimated by using the inventory-output ratios for input-outputindustries for 1963 in US Department of Commerce (1973).
We required a 71-order price-index vector, e, to factor out changes in input-output coefficientswhich are due to market price changes. This is necessary if we are to compare technologies over time,since we are using a 'dollar's worth' as a measure of physical quantity of a product (see Carter, 1970,p. 21). The US Department of Labor (1979) provides price indices for most input-output industries.Except for 1947, all the required indices are included in the data tape of Research Data Associates(1982). We used the price index vector implicit in US Department of Commerce (1970)—whichgives 1947 input-output data in 1947 and 1958 dollars—to obtain the 1947 values.
Summarising, we have
A = Input-output coefficients matrix;k = Fixed-capital coefficients row vector;&o = Direct reduced-labour coefficients row vector;w <= Capital-assets weights matrix;G = Gross-capital-stocks matrix;b' = Real-wage column vector per unit of reduced labour;1 = Asset-lifetime column vector (per asset type);e = Price-index row vector;q = Dollar value of output-by-industry row vector;K = Capital-stocks coefficients matrix;
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Values, prices, and wage-profit curves in the US economy 429D = Depreciation coefficients matrix;H = Gross-investment coefficients matrix;g = Indirect-tax coefficients row vector;t = Turnover-time row vector.
Given this data base, we could perform all the computations outlined in the paper. In all cases, wecan compute prices and values using the data in current dollars or in constant dollars. The former ismore accurate when investigating relations in a single year; the latter is necessary for intertemporalcomparisons.
To deflate the A matrix we computed:
A*=<e->>A<e> (A5)
where < e > is the diagonal matrix obtained from the vector e.To deflate the output row vector q:
q* = q<«"'> (A6)
To handle the capital valuation problem, we took the capital coefficients matrix K at currentdollars and deflated it like A. This is valid since current dollar gross-stocks measure currentreplacement value of assets still in place.
The real unit-wage vector b, also measured in 'dollar's worth', was deflated as q above.
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